Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW)
Citation:
H.A. Lorentz, The motion of electrons in metallic bodies II, in:
KNAW, Proceedings, 7, 1904-1905, Amsterdam, 1905, pp. 585-593
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-1-
( 585 )
this experiment, not with garden soil for infection material, but by
using the stomacal contents of sueh a case of stomaeal sarcine. The
"not cultivability" of DE BARY may mean the same as anaerobiosis,
fol' it is weU known how difiicult it is, even at the present time,
to eultivate anaerobics if the particulal's of their life conditions are
not exactly known.
For the rest I do not doubt of the pl'ecision OfJhLUNHEIM'S 1) and
MIGULA'S 2) obsel'vations, wno have seen aerobic colonies of microcoeci
originate from stomacnJ sar('ine. It is true that 1 fol' my part. have
not succeeded in confirroing this observation with regal'd to the
fermentation &arcine, but fol' other speeies of 8a1'cina I have, with
certainty, stated the transition into micrococci, and with various
anaerobics, although not belonging to the genus Sa1'cina, I have seen
nowand then colonies originate of facllltative anael'obics, which in all
othel' l'espects, eOl'responded to the obligative anaerobics used for the
cultures. Therefore this modification also seems possible for some
individuals of the fermentation sal'cine. Aecumulation Or transfer
experiment& with stomacal contents will howevel' only then give
positive results, if these are used when still in fermentation; with
long kept material nothing can be expected.
A.lready the older obsel'vers 3) as SCHLOSSBERGER (1847), SIMON
(1849) and CRAMER (1858) have tried, altbough in vaiu, by a kind of
accumulation expel'iments, fo cuitivate the stomacal sarcine, wherefore
they pl'epared, as nutl'ient liquid, artificial gastrie juice with different
additions. Remarkable, and illustrating the biological views of
those days, is the fact, that for the infection they did not use
the stomacal contents themselves, but beer yeast, supposing, that the
sarcine might originate ti'om fhe yeast eeUs, which somewhat l'esemble
it, and are always tound in the stomach togethel' with the sarcine
itself.
"Tlte motion of electl'ons in metallic bodies." Il. By
Physics. -
Prof. H. A..
LORENTZ.
(Communicated in the meeting of January 28, 1905).
~
11. By a mode of reasoning similar to that used in the last ~,
we may deduce a formula tOl' the intensity i of the eurl'ent in a
closed thel'mo-eleetric circuit. For this pUl'pose we haye only to
suppose tbe ends Pand Q, which consist, as has been said, of the
1) Archiv f. experiment. Pathologie und Pharmacologie. Bd. IO,
2) System der Bacteriën. Bd. 2, pg. 259, 1900.
S) Cited from
SURINGAR
pg. 339, 188:'.
(I. c.).
40
Proceedings Roya! Acad. Amsterdam. Vol. VII.
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same metal and are kept at the same temperature, to be brought in
contact with each othel'. The potentials rpp and po, will then become
equa!, but the stl'eam of electl'ons v will no longel' be O. We shall
have on the contl'al'y, denoting' by :2 the nOl'mal section, which may
slowly change from point to point, as has already been observed,
=
i
e v:2
"
(36)
Taking this into account and using (23), we get ti'om (21) and (30)
dp
dre
1
dV
= - -; dal -
m d
-; d,v
(1)h -
m d log A l .
2elt -----;;:;;- - a:2 z.
We sha11 integrate this along the circuit from P to Q. Since i
has the same value evel'ywhere and
Pp
=
Po, , V p
we find
=
V o,
q.
_ m f~ d log A dre _
2e h dm
p
lt p
=
lt o"
I
ija:2 = o.
dre
Here, the first term is redllced to the form (34), if we integrate
by parts. Hence, if we put
the result is
J:;=R,
F=iR, R'
F
i=
as was to be expected. lndeed, a being the coefficient of conductivity,
R is the resistance of the circuit.
I
§ 12. We shall now proceed to calculate the heat developed in a
circuit iiJ. which th ere is an electric CUlTent i, 01' rather, supposing
each element of the wire to be kept at a constant temperatlll'e by
means of an extern al reservoir of heat, the amount of heat th at is
transferred to such a reservoir pel' unit time. Let us consider to this
effect the part of the circuit lying between the sections whose positions
daJ and let u'dt be the work done,
are determined by aJ and aJ
during the time dt, by the forces acting on the eJectrons in this
element. W:2 being the q uantity of heat travel'sing a section pel'
unit time, we may write
+
d
dm (W :2) dm
for the diffel'ence between the quantities of heat leaving the element
at one end and entering it at the othel', a,nd the prodllction of hea,t
is given by
-3-
( 587 )
q
d
,..,
(W .l.)d.lJ.
dIJ)
=w -
-
.
.
. .
. .
(37)
In order to determine 'tV, we obsel've in the first place that the'
wOl'k done, dUl'ing the time dt, by the force acting on a single
electron is
mXgdt
and that, by the fOl'mula (1), \ the element :Sda; contains
fes,
1],~):2
dIJ} dl
electl'ons having theil' velocity-points within the element dl of the
diagram of velocities. Taking togethel' the farces acting on all these
pal'tip,les, we find fol' theil' work
m X ~ dIJ) dt . gf(g, 1], ~) dl ,
an expression th at has yet to be integrated over the whole extent
of the diagram. On account of (4), the result becomes
mXv2d.vdt,
sa that, by (36)
miX
w=-dlJ).
e
Now, the value of X may be taken from (21).
Substituting
2
v=e2
and using at the same time (23), we find
X
=
1----a:;;- + (1)h +
dlogA
2h
d
dal
ei
mfJ2'
.
• • . (38)
sa that
if we put
[1
d log
A
w1 -- mi
-e -2/t d.v-
+ a.v
-d (1)]
-A dre• •
• • • • (39)
and
i'
w, :::: d.v.
fJ2
The expl'ession (22) may likewise be transformed by introducing
into it the value (38), Ol', what amounts to the same thing, the
value of
dA
21tAX--,
d.v
tl}at may be drawn fl'om (21). One finds in this way
W=W1+W,.
-4-
( 588 )
if
Wl
2:rmlA dit
3lt
dtv
=--4
.......
(40)
and
m
mi
. . . . . . (41)
h
e2lt
§ 13. The expression (37) fol' the amount of heat produced in
the element dx may now be dlvided into three parts.
The th'st of these
W.=-v=-.
i'
I
w,:=:-dm
I
62
corresponds to
JOULE'S
law. Indeed
dtv
62
of the circuit extending ti'om (x) to (x
The second part
d
dtlJ
is the l'esistance of the part
+ dx).
--(W 2) dm
1
is entirely independent of the current, as appears f,'om (40). It may
therefore be considered to be due to ol'dinal'y ronduction of heat.
This is confirmed by comparing it with what bas been said in § 9.
It remains to consider the quantity of heat
=
qI
Wl -
d ( W2 2) dtv,
d.1!
or, if (39) and (41) are taken into, account,
I
mi dlogA
q=---d.'IJ.
2 e lt dtlJ
This expre&sion, proportional to the CUl'rent and rhanging its sign
if 1he latter is reversed, will lead uS to formulae fol' the PELTlER.
effect and the THoMsoN-effect. Reduced to unit current, it becomes
,
q
1=1
m dlogA
=2elt
- --d.'IJ
d.7·
.
"
.
(42)
a. 1 shall suppose in the first pI ace that, between two sections
of the circuit, there is a gradual transition from the metal 1 to the
metal H, the tempel'ature and consequeIltly!t being the same thl'oughout t11is part of the circuit. Then, reckoning ,'IJ from the metal 1
towal'ds 1I, and integrating l42), 1 Hnd fol' the heat produced at the
"plare of contact" by a cm'rent of unit strength flowing from 1
towal'ds I1,
ml (All) _
_ 2al'
- - log (All)
.
3 e . Al
og 2 e lt ' A l
-5-
( 5M9 )
Hence, if we cha,racterize the PEJ,TIER-effect by the ab,,01'1Jtion of
heat 11/,11 taking place in this case,
lI/,1l
ó. In
we shaH
then to
we may
2 a l'
= Te
log
(Al) ,= s-;-l' (NI)
2a
AII
Nn
log
.
.
.
(43)
the second place, substituting again for h the value (14),
a,pply (42) to a homogeneous part of the circuit. We have
conslder log A as a function of the temperature T, so thnt
write
2aTdlogA
, =- -dT
-q 1=1
3e
dl'
fOl" the heat deyeloped between two points !rept at the temperatnres
T a,nd T cl T, if a current of unit strellgth flows from the first
point towards the latter. What KELVIN has caUed the "spccific heat
of electricity" (THOl\ISoN·effect) is thu5 seen to be represented by
+
2aTdlogA
(.L=---- . . . . . . .
3e
dT
(44)
§ 14. An important feature of the abm"e results is their agreement
with those of the weU known thermodynamic theory of thermoelectric currents. This theory leads /0 the relations
and
.
.
(45)
.
.
(46)
Til
F= -
Ji:
IJ
dT,
.
.
.
.
.
T'
in which
and (.Lll are the specific heats of electricity in the
metals land Il, at the temperature '1', whereas F denotes what we
have calculated in § 10, viz. the electromotive force in a circuit
composed of these metals and whose junctions are kept at the
temperatures T' a,nd T/I, the force being reckoned positive if it
tends to produce a, current which flo\vs ti'om I towards II through
the fh'st junction.
The values (44), (43) and (35) are easily seen to satisfy the
equations (45) and (46).
Instead of verifying this, we may as weIl infer directly from (42)
that our results agree with what is l'equired by the laws of thermodynamics. On account of the first of these we must have
(.11
:Eq'/=l= -F
and by the second
-6-
J
590 )
'.
:2 q1-1
0
1
'
the sums in these formulae relating to all elements of -the closed
circllit we have examined in § j 1. Now, by (42), these formulae
become
=
Q
F
= _ mJ~ ~lOfl A dm
2e
p
lt
dtv
and
Q
J
I
1 dlogA
'tT
dm
dtv
= O.
p
The first of these equations is identical with (34:) and the second
holds because kT has evel'ywhel'e the same value.
It must also be noticed that the fOl'mula (35) implies the existence
of a thermo-electl'ic series and the weIl known law l'elating to it.
This follows at once from the fact that the value (35) may be
written as the difference of two integrals depending, for given
temperatures of the junctions, the one on the properties of the fil'st
and the other on those of the second meta!. Denoting by III a third
metal, we may represent by Fl, 11, FlI, 111, FIlI, 1 the electromotive
forces existing in circuits composed of the metals indicated by the
indices, the junctions having in all these cases the temperatUl'es T'
and Til and the positive direction being such that it leads thl'ough
the junction at the fi1'st tempeI'ature from the metal indicttted in
the first towa1'ds that indicated in the second pl~ce. Then it iil seen
at once that
FE, IJ
FIl, III
PlIl, [ = O. . . . . . (47)
Strictly speaking there was no need to prove this, as it is a consequenee of the thermody:r;tamic equations and our l'esults ag1'ee
with these.
+
+
§ 15. In what pl'ecedes we have assumed a single kind of free
eJectrons. Indeed, many observations on other classes of phenomena
have shown the negative electrons to llave a gl'eate1' mobility than
the positive ones, so that one feels inclined to ask in the first place
to what extent the faets may be explained by a theo1'Y wo1'king
with only negative free eItlCtrons.
Now, in el:amining this point, we have first of all to eonside~' the
absolute value of the eleetl'omotive force F. It' we suppose the tempel'atures 1" /lnd Til to diffel' hy one degl'ee and if we ~neglect thc
I
-7-
( 591 )
variability of Ni and NlI in so small an interval, we may write
for (35)
20.
Nn
Flo
log
=3; log NI'
NlI '3e
- = - , Flo,
Nr
20.
The value of the first factor -on the righthand side may be taken
from what, in § 9, we have deduced from the electl'ochemical
aequivalent of hydrogen 1). We found fol' T
291
=
aT
-=38x10 5,
e
that
50
NII
log Nr
= 0,00011]i~
1°'
In the case of bismuth and antimony, Flo amounts to 12000,
corresponding to
NJ[
NIl_ 37
log-= 1,32
NI - ,
NI
I see no ditlicult,v in admitting this mtio bet ween the mImber of
free electl'ons in two metals wide apart from each other in the
thermo-electric series 2).
1) The numbers of that § cODtain an error which, however, has no influence
on the agreement that ~hould be established by them. The value of 3 pand that of
oc1.1
- deduced from the measurements of JAEGER and DIESSELHORST are not 38 and
e
47, but
3p
= 38 X
10~
and
aT =47
e
X ]0 5 ,
Nu
2) Let n be the mean value of log NI between tbe tempel'atuz'es T' and T'/,
Then the equation (35) may be put in the fOl'ln
2
Pe = - na (1''' - 1").
3
This may be expressed as folJows: The work done by the electromotive force
in case one electron travels al'ound tbe circuit is found if we multiply by
~n
the increase of the meau kinetic enèl'gy of a gaseous molecule, due 10 an elevation
of tempel'ature from T' to T",
-8-
( 592 )
'rhe question now arises whether it wiIl be possible to explain
all observations in the domain of thermo-electricity by roeans of
suitable assumptions concerning the number of free electrons. In order
to form an opinion on tl1is point, I shaU suppose the PELTIER-effect
to be known, at one definite temperature T o, for all combinations
of some standard metal with other metals and the THoMsoN-effect to
have been measured in all metals at all temperatures. 'l'hen, aftel'
having chosen arbitrarily the number No of free electrons in the
standard metal at T o' we may deduce from (43) the corresponding
values for the other conductors, and the eql1ation (44) combined with
(13) and l14), will serve to determine, for all metals, the value of
N at any temperature we like. Now, the numbers obtained in this
way, all of which contain No as an indeterminate factor, will sufiice
to account for all other thermo-electric phenomena. at least if we
take for granted that these phenomena obey the Jaws deduced from
thermodynamics. Indeed, these Jaws Jeading to the l'elation
11], 11
L
+ IIl1. III + IIl1I. I = 0,
similar to (47), the values of N we have assumed will account not
only for the PELTIER-effect at the temperature T o for all metals
combined with the standal'd metal, but also for the effect, at the
same tempel'atnre, for any combination. Finally, we se€.' fi'om (45)
that the value of nr.lI at any ternperature may be found from that
corresponding to T o' if we know the THoMsoN-effect for all intermediate temperatures and from (46) th at the values of the electromotive force are determined by those of II.
There is but one difficulty that might arise in this comparison I of
theory with experimental l'esults; it might be that the assumptions
we should have to make concerning the numbers N would prove
incompatible with theoretical considel'ations of one kind or another
about the causes which determine tIle number of free electl'ons.
As to the conductivities fol' heat and electl'icity, it would always
he possible to obtain the right values ti'om (24) and (27), provided
only we make appropl'iate assnmptions concerning tbe length l of
the free path between two encounters 1).
It must be noticed, however, that, whatever be the value of tbis
length, tlle fOl'egoing theory requires that the ratio
!:...
()
shall be the
1) lf the elect! ie conduclivlty were mvelsely plOpOl tiollal to the absolute temperature, a'5 it is applm..imately for some metals, alld if we might neglect the variations of N, the lormula (24) would requil'e that t is iuverselY pl'oportional lo
VT. I am unable to explain why N should val'y iu this way.
-9-
t
593 )
same fol' all metals. The rather large deviations from this law have
led DUUDE to assnme more than alle kind of free electrans, au hypothesis we shall have to discuss in a &equel to this paper. For the
moment I shaH only obsel've that one reason fol' admitting the
existence n01 onIy of negative but aIso of positive free eIectrons
lies in the fact that the HALL-effect has not in all metals the same
dil'ection.
(March 22, 1905).
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